Computing endomorphism rings of Jacobians of genus 2 curves over finite fields
| dc.creator | Freeman, David | |
| dc.creator | Lauter, Kristin | |
| dc.date | 2007-01-10 | |
| dc.date | 2007-05-30 | |
| dc.date.accessioned | 2026-07-07T08:08:36Z | |
| dc.date.available | 2026-07-07T08:08:36Z | |
| dc.description | We present algorithms which, given a genus 2 curve $C$ defined over a finite field and a quartic CM field $K$, determine whether the endomorphism ring of the Jacobian $J$ of $C$ is the full ring of integers in $K$. In particular, we present probabilistic algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups $J[\ell^d]$ for prime powers $\ell^d$. We use these algorithms to create the first implementation of Eisenträger and Lauter's algorithm for computing Igusa class polynomials via the Chinese Remainder Theorem \cite{el}, and we demonstrate the algorithm for a few small examples. We observe that in practice the running time of the CRT algorithm is dominated not by the endomorphism ring computation but rather by the need to compute $p^3$ curves for many small primes $p$. | |
| dc.description | Revised version, with minor corrections and incorporating reader comments. Proposition 3.7 and Lemma 6.5 are new. To appear in Proceedings of SAGA 2007, Tahiti | |
| dc.identifier | https://arxiv.org/abs/math/0701305 | |
| dc.identifier | http://arxiv.org/abs/math/0701305 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/131317 | |
| dc.subject | Number Theory | |
| dc.subject | 11G25; 11G15 (Primary); 14G50 (Secondary) | |
| dc.title | Computing endomorphism rings of Jacobians of genus 2 curves over finite fields | |
| dc.type | text |